Schutz B. A first course in general relativity
Подождите немного. Документ загружается.
ix Contents
t
7.4 Conserved quantities 178
7.5 Further reading 181
7.6 Exercises 181
8 The Einstein field equations 184
8.1 Purpose and justification of the field equations 184
8.2 Einstein’s equations 187
8.3 Einstein’s equations for weak gravitational fields 189
8.4 Newtonian gravitational fields 194
8.5 Further reading 197
8.6 Exercises 198
9 Gravitational radiation 203
9.1 The propagation of gravitational waves 203
9.2 The detection of gravitational waves 213
9.3 The generation of gravitational waves 227
9.4 The energy carried away by gravitational waves 234
9.5 Astrophysical sources of gravitational waves 242
9.6 Further reading 247
9.7 Exercises 248
10 Spherical solutions for stars 256
10.1 Coordinates for spherically symmetric spacetimes 256
10.2 Static spherically symmetric spacetimes 258
10.3 Static perfect fluid Einstein equations 260
10.4 The exterior geometry 262
10.5 The interior structure of the star 263
10.6 Exact interior solutions 266
10.7 Realistic stars and gravitational collapse 269
10.8 Further reading 276
10.9 Exercises 277
11 Schwarzschild geometry and black holes 281
11.1 Trajectories in the Schwarzschild spacetime 281
11.2 Nature of the surface r = 2M 298
11.3 General black holes 304
11.4 Real black holes in astronomy 318
11.5 Quantum mechanical emission of radiation by black holes:
the Hawking process 323
11.6 Further reading 327
11.7 Exercises 328
12 Cosmology 335
12.1 What is cosmology? 335
12.2 Cosmological kinematics: observing the expanding universe 337
x Contents
t
12.3 Cosmological dynamics: understanding the expanding universe 353
12.4 Physical cosmology: the evolution of the universe we observe 361
12.5 Further reading 369
12.6 Exercises 370
Appendix A Summary of linear algebra 374
References 378
Index 386
Preface to the second edition
In the 23 years between the first edition of this textbook and the present revision, the field
of general relativity has blossomed and matured. Upon its solid mathematical foundations
have grown a host of applications, some of which were not even imagined in 1985 when
the first edition appeared. The study of general relativity has therefore moved from the
periphery to the core of the education of a professional theoretical physicist, and more and
more undergraduates expect to learn at least the basics of general relativity before they
graduate.
My readers have been patient. Students have continued to use the first edition of this
book to learn about the mathematical foundations of general relativity, even though it has
become seriously out of date on applications such as the astrophysics of black holes, the
detection of gravitational waves, and the exploration of the universe. This extensively
revised second edition will, I hope, finally bring the book back into balance and give
readers a consistent and unified introduction to modern research in classical gravitation.
The first eight chapters have seen little change. Recent references for further reading
have been included, and a few sections have been expanded, but in general the geometrical
approach to the mathematical foundations of the theory seems to have stood the test of time.
By contrast, the final four chapters, which deal with general relativity in the astrophysical
arena, have been updated, expanded, and in some cases completely re-written.
In Ch. 9, on gravitational radiation, there is now an extensive discussion of detection
with interferometers such as LIGO and the planned space-based detector LISA. I have
also included a discussion of likely gravitational wave sources, and what we can expect
to learn from detections. This is a field that is rapidly changing, and the first-ever direct
detection could come at any time. Chapter 9 is intended to provide a durable framework
for understanding the implications of these detections.
In Ch. 10, the discussion of the structure of spherical stars remains robust, but I have
inserted material on real neutron stars, which we see as pulsars and which are potential
sources of detectable gravitational waves.
Chapter 11, on black holes, has also gained extensive material about the astrophysical
evidence for black holes, both for stellar-mass black holes and for the supermassive black
holes that astronomers have astonishingly discovered in the centers of most galaxies. The
discussion of the Hawking radiation has also been slightly amended.
Finally, Ch. 12 on cosmology is completely rewritten. In the first edition I essentially
ignored the cosmological constant. In this I followed the prejudice of the time, which
assumed that the expansion of the universe was slowing down, even though it had not yet
been accurately enough measured. We now believe, from a variety of mutually consistent
observations, that the expansion is accelerating. This is probably the biggest challenge to
xii Preface to the second edition
t
theoretical physics today, having an impact as great on fundamental theories of particle
physics as on cosmological questions. I have organized Ch. 12 around this perspective,
developing mathematical models of an expanding universe that include the cosmological
constant, then discussing in detail how astronomers measure the kinematics of the universe,
and finally exploring the way that the physical constituents of the universe evolved after the
Big Bang. The roles of inflation, of dark matter, and of dark energy all affect the structure
of the universe today, and even our very existence. In this chapter it is possible only to give
a brief taste of what astronomers have learned about these issues, but I hope it is enough to
encourage readers to go on to learn more.
I have included more exercises in various chapters, where it was appropriate, but I have
removed the exercise solutions from the book. They are available now on the website for
the book.
The subject of this book remains classical general relativity; apart from a brief discussion
of the Hawking radiation, there is no reference to quantization effects. While quantum
gravity is one of the most active areas of research in theoretical physics today, there is still
no clear direction to point a student who wants to learn how to quantize gravity. Perhaps
by the third edition it will be possible to include a chapter on how gravity is quantized!
I want to thank many people who have helped me with this second edition. Several have
generously supplied me with lists of misprints and errors in the first edition; I especially
want to mention Frode Appel, Robert D’Alessandro, J. A. D. Ewart, Steve Fulling, Toshi
Futamase, Ted Jacobson, Gerald Quinlan, and B. Sathyaprakash. Any remaining errors are,
of course, my own responsibility. I thank also my editors at Cambridge University Press,
Rufus Neal, Simon Capelin, and Lindsay Barnes, for their patience and encouragement.
And of course I am deeply indebted to my wife Sian for her generous patience during all
the hours, days, and weeks I spent working on this revision.
Preface to the first edition
This book has evolved from lecture notes for a full-year undergraduate course in general
relativity which I taught from 1975 to 1980, an experience which firmly convinced me
that general relativity is not significantly more difficult for undergraduates to learn than
the standard undergraduate-level treatments of electromagnetism and quantum mechanics.
The explosion of research interest in general relativity in the past 20 years, largely stimu-
lated by astronomy, has not only led to a deeper and more complete understanding of the
theory, it has also taught us simpler, more physical ways of understanding it. Relativity is
now in the mainstream of physics and astronomy, so that no theoretical physicist can be
regarded as broadly educated without some training in the subject. The formidable rep-
utation relativity acquired in its early years (Interviewer: ‘Professor Eddington, is it true
that only three people in the world understand Einstein’s theory?’ Eddington: ‘Who is the
third?’) is today perhaps the chief obstacle that prevents it being more widely taught to
theoretical physicists. The aim of this textbook is to present general relativity at a level
appropriate for undergraduates, so that the student will understand the basic physical con-
cepts and their experimental implications, will be able to solve elementary problems, and
will be well prepared for the more advanced texts on the subject.
In pursuing this aim, I have tried to satisfy two competing criteria: first, to assume a min-
imum of prerequisites; and, second, to avoid watering down the subject matter. Unlike most
introductory texts, this one does not assume that the student has already studied electro-
magnetism in its manifestly relativistic formulation, the theory of electromagnetic waves,
or fluid dynamics. The necessary fluid dynamics is developed in the relevant chapters. The
main consequence of not assuming a familiarity with electromagnetic waves is that grav-
itational waves have to be introduced slowly: the wave equation is studied from scratch.
A full list of prerequisites appears below.
The second guiding principle, that of not watering down the treatment, is very subjective
and rather more difficult to describe. I have tried to introduce differential geometry fully,
not being content to rely only on analogies with curved surfaces, but I have left out subjects
that are not essential to general relativity at this level, such as nonmetric manifold theory,
Lie derivatives, and fiber bundles.
1
I have introduced the full nonlinear field equations,
not just those of linearized theory, but I solve them only in the plane and spherical cases,
quoting and examining, in addition, the Kerr solution. I study gravitational waves mainly
in the linear approximation, but go slightly beyond it to derive the energy in the waves
and the reaction effects in the wave emitter. I have tried in each topic to s upply enough
1
The treatment here is therefore different in spirit from that in my book Geometrical Methods of Mathematical
Physics (Cambridge University Press 1980b), which may be used to supplement this one.
xiv Preface to the first edition
t
foundation for the student to be able to go to more advanced treatments without having to
start over again at the beginning.
The first part of the book, up to Ch. 8, introduces the theory in a sequence that is typi-
cal of many treatments: a review of special relativity, development of tensor analysis and
continuum physics in special relativity, study of tensor calculus in curvilinear coordinates
in Euclidean and Minkowski spaces, geometry of curved manifolds, physics in a curved
spacetime, and finally the field equations. The remaining four chapters study a few top-
ics that I have chosen because of their importance in modern astrophysics. The chapter
on gravitational radiation is more detailed than usual at this level because the observa-
tion of gravitational waves may be one of the most significant developments in astronomy
in the next decade. The chapter on spherical stars includes, besides the usual material, a
useful family of exact compressible solutions due to Buchdahl. A long chapter on black
holes studies in some detail the physical nature of the horizon, going as far as the Kruskal
coordinates, then exploring the rotating (Kerr) black hole, and concluding with a simple
discussion of the Hawking effect, the quantum mechanical emission of radiation by black
holes. The concluding chapter on cosmology derives the homogeneous and isotropic met-
rics and briefly studies the physics of cosmological observation and evolution. There is an
appendix summarizing the linear algebra needed in the text, and another appendix contain-
ing hints and solutions for selected exercises. One subject I have decided not to give as
much prominence to, as have other texts traditionally, is experimental tests of general rel-
ativity and of alternative theories of gravity. Points of contact with experiment are treated
as they arise, but systematic discussions of tests now require whole books (Will 1981).
2
Physicists today have far more confidence in the validity of general relativity than they had
a decade or two ago, and I believe that an extensive discussion of alternative theories is
therefore almost as out of place in a modern elementary text on gravity as it would be in
one on electromagnetism.
The student is assumed already to have studied: special relativity, including the Lorentz
transformation and relativistic mechanics; Euclidean vector calculus; ordinary and simple
partial differential equations; thermodynamics and hydrostatics; Newtonian gravity (sim-
ple stellar structure would be useful but not essential); and enough elementary quantum
mechanics to know what a photon is.
The notation and conventions are essentially the same as in Misner et al., Gravitation
(W. H. Freeman 1973), which may be regarded as one possible follow-on text after this one.
The physical point of view and development of the subject are also inevitably influenced
by that book, partly because Thorne was my teacher and partly because Gravitation has
become such an influential text. But because I have tried to make the subject accessible
to a much wider audience, the style and pedagogical method of the present book are very
different.
Regarding the use of the book, it is designed to be studied sequentially as a whole, in
a one-year course, but it can be shortened to accommodate a half-year course. Half-year
courses probably should aim at restricted goals. For example, it would be reasonable to aim
to teach gravitational waves and black holes in half a year to students who have already
2
The revised second edition of this classic work is Will (1993).
xv Preface to the first edition
t
studied electromagnetic waves, by carefully skipping some of Chs. 1–3 and most of Chs. 4,
7, and 10. Students with preparation in special relativity and fluid dynamics could learn
stellar structure and cosmology in half a year, provided they could go quickly through the
first four chapters and then skip Chs. 9 and 11. A graduate-level course can, of course, go
much more quickly, and it should be possible to cover the whole text in half a year.
Each chapter is followed by a set of exercises, which range from trivial ones (filling
in missing steps in the body of the text, manipulating newly introduced mathematics) to
advanced problems that considerably extend the discussion in the text. Some problems
require programmable calculators or computers. I cannot overstress the importance of
doing a selection of problems. The easy and medium-hard ones in the early chapters give
essential practice, without which the later chapters will be much less comprehensible. The
medium-hard and hard problems of the later chapters are a test of the student’s understand-
ing. It is all too common in relativity for students to find the conceptual framework so
interesting that they relegate problem solving to second place. Such a separation is false
and dangerous: a student who can’t solve problems of reasonable difficulty doesn’t really
understand the concepts of the theory either. There are generally more problems than one
would expect a student to solve; several chapters have more than 30. The teacher will
have to select them judiciously. Another rich source of problems is the Problem Book in
Relativity and Gravitation, Lightman et al. (Princeton University Press 1975).
I am indebted to many people for their help, direct and indirect, with this book. I would
like especially to thank my undergraduates at University College, Cardiff, whose enthu-
siasm for the subject and whose patience with the inadequacies of the early lecture notes
encouraged me to turn them into a book. And I am certainly grateful to Suzanne Ball, Jane
Owen, Margaret Vallender, Pranoat Priesmeyer, and Shirley Kemp for their patient typing
and retyping of the successive drafts.
1
Special relativity
1.1 Fundamental principles of special relativity (SR)
theory
The way in which special relativity is taught at an elementary undergraduate level – the
level at which the reader is assumed competent – is usually close in spirit to the way it was
first understood by physicists. This is an algebraic approach, based on the Lorentz transfor-
mation (§ 1.7 below). At this basic level, we learn how to use the Lorentz transformation to
convert between one observer’s measurements and another’s, to verify and understand such
remarkable phenomena as time dilation and Lorentz contraction, and to make elementary
calculations of the conversion of mass into energy.
This purely algebraic point of view began to change, to widen, less than four years
after Einstein proposed the theory.
1
Minkowski pointed out that it is very helpful to regard
(t, x, y, z) as simply four coordinates in a four-dimensional space which we now call space-
time. This was the beginning of the geometrical point of view, which led directly to general
relativity in 1914–16. It is this geometrical point of view on special relativity which we
must study before all else.
As we shall see, special relativity can be deduced from two fundamental postulates:
(1) Principle of relativity (Galileo): No experiment can measure the absolute velocity of
an observer; the results of any experiment performed by an observer do not depend on
his speed relative to other observers who are not involved in the experiment.
(2) Universality of the speed of light (Einstein): The speed of light relative to any unac-
celerated observer is c = 3 ×10
8
ms
−1
, regardless of the motion of the light’s source
relative to the observer. Let us be quite clear about this postulate’s meaning: two differ-
ent unaccelerated observers measuring the speed of the same photon will each find it to
be moving at 3 ×10
8
ms
−1
relative to themselves, regardless of their state of motion
relative to each other.
As noted above, the principle of relativity is not at all a modern concept; it goes back
all the way to Galileo’s hypothesis that a body in a state of uniform motion remains in that
state unless acted upon by some external agency. It is fully embodied in Newton’s second
1
Einstein’s original paper was published in 1905, while Minkowski’s discussion of the geometry of spacetime
was given in 1908. Einstein’s and Minkowski’s papers are reprinted (in English translation) in The Principle of
Relativity by A. Einstein, H. A. Lorentz, H. Minkowski, and H. Weyl (Dover).
2 Special relativity
t
law, which contains only accelerations, not velocities themselves. Newton’s laws are, in
fact, all invariant under the replacement
v(t) → v
(t) = v(t) −V,
where V is any constant velocity. This equation says that a velocity v(t) relative to one
observer becomes v
(t) when measured by a second observer whose velocity relative to the
first is V. This is called the Galilean law of addition of velocities.
By saying that Newton’s laws are invariant under the Galilean law of addition of veloc-
ities, we are making a statement of a sort we will often make in our study of relativity,
so it is well to start by making it very precise. Newton’s first law, that a body moves at a
constant velocity in the absence of external forces, is unaffected by the replacement above,
since if v(t) is really a constant, say v
0
, then the new velocity v
0
− V is also a constant.
Newton’s second law
F = ma = m dv/d t,
is also unaffected, since
a
− dv
/d t = d(v −V)/d t = dv/d t = a.
Therefore, the second law will be valid according to the measurements of both observers,
provided that we add to the Galilean transformation law the statement that F and m are
themselves invariant, i.e. the same regardless of which of the two observers measures them.
Newton’s third law, that the force exerted by one body on another is equal and opposite to
that exerted by the second on the first, is clearly unaffected by the change of observers,
again because we assume the forces to be invariant.
So there is no absolute velocity. Is there an absolute acceleration? Newton argued that
there was. Suppose, for example, that I am in a train on a perfectly smooth track,
2
eating a
bowl of soup in the dining car. Then, if the train moves at constant speed, the soup remains
level, thereby offering me no information about what my speed is. But, if the train changes
its speed, then the soup climbs up one side of the bowl, and I can tell by looking at it how
large and in what direction the acceleration is.
3
Therefore, it is reasonable and useful to single out a class of preferred observers: those
who are unaccelerated. They are called inertial observers, and each one has a constant
velocity with respect to any other one. These inertial observers are fundamental in spe-
cial relativity, and when we use the term ‘observer’ from now on we will mean an inertial
observer.
The postulate of the universality of the speed of light was Einstein’s great and radical
contribution to relativity. It smashes the Galilean law of addition of velocities because it
says that if v has magnitude c, then so does v
, regardless of V. The earliest direct evidence
for this postulate was the Michelson–Morely experiment, although it is not clear whether
Einstein himself was influenced by it. The counter-intuitive predictions of special relativity
all flow from this postulate, and they are amply confirmed by experiment. In fact it is
probably fair to say that special relativity has a firmer experimental basis than any other of
2
Physicists frequently have to make such idealizations, which often are far removed from common experience !
3
For Newton’s discussion of this point, see the excerpt from his Principia in Williams (1968).